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Chapter 14 – Game Theory

14.1 Nash Equilibrium14.2 Repeated Prisoners’ Dilemma

14.3 Sequential-Move Games and

Strategic Moves

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Game Theory and LifeYou are on a first date with the love of

your dreams. You can propose 2 activities:

1)Safe activity (Coffee)2)Exciting Activity (Waterpark)

Your date could either want a safe activity or an exciting activity. There are different results if your ideas match up or clash:

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Mr/MissRightMr/MissRight

YouYou

Chapter Fourteen

First Date GameWhat is the outcome of this game?Payoff format is (Left, Top)

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Game Theory ComponentsPlayers: agents participating in the game (You and Your Date

Strategies: Actions that each player may take under any possible circumstance (Coffee, Waterpark)

Outcomes: The various possible results of the game (four, each represented by one cell of the payoff matrix)

Payoffs: The benefit that each player gets from each possible outcome of the game (the profits entered in each cell of the payoff matrix)

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Best ResponsesIn all game theory games, players choose

strategies without knowing with certainty what the opposing player will do.

Players construct BEST RESPONSES

-optimal actions given all possible actions of other players

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Mr/MissRightMr/MissRight

YouYou

Chapter Fourteen

First Date Game Best ResponsesIf you know your date will pick coffee, you should pick coffee, since 10 > -5If you know your date will pick waterpark, you should pick waterpark, since 20 > 0

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Mr/MissRightMr/MissRight

YouYou

Chapter Fourteen

First Date Game Best ResponsesIf your date knows you will pick coffee, they should pick coffee, since 10 > -5If your date knows you will pick waterpark, they should pick waterpark, since 20 > 0

Note that this game is SYMMETRICAL

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Nash Equilibrium

Definition: A Nash Equilibrium occurs when each player chooses a strategy that gives him/her the highest payoff, given the strategy chosen by the other player(s) in the game. ("rational self-interest")

Nash Equilibria occur when best responses line up

The Date Game:

Nash equilibria: Each proposes coffee or each proposes waterpark.

Definition: A Nash Equilibrium occurs when each player chooses a strategy that gives him/her the highest payoff, given the strategy chosen by the other player(s) in the game. ("rational self-interest")

Nash Equilibria occur when best responses line up

The Date Game:

Nash equilibria: Each proposes coffee or each proposes waterpark.

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Game Theory

•A special kind of Best Response:

•Strategy that is best no matter what the other player does.

DOMINANT STRATEGY

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Advertising

A’s profit= $50 000

A’s loss =

$25 000

A’s profit= $75 000

A’s profit = $10 000

B’s profit = $50 000

B’s profit = $75 000

B’s loss = $25 000

B’s profit = $10 000

Don’t advertise AdvertiseB’s

STRATEGY

A’s STRATEGY

Don’t advertise

Advertise

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Dominant Strategy

A’s profit= $50 000

A’s loss =

$25 000

A’s profit= $75 000

A’s profit = $10 000

B’s profit = $50 000

B’s profit = $75 000

B’s loss = $25 000

B’s profit = $10 000

Don’t advertise AdvertiseB’s

dominant strategy is advertise

A’s dominant strategy is advertise

Don’t advertise

Advertise

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Prisoner’s Dilemma

• This is an example of a prisoner’s dilemma type of game.– There is dominant strategy. – The dominant strategy does not result in the best

outcome for either player.– It is hard to cooperate even when it would be

beneficial for both players to do so– Cooperation between players is difficult to maintain

because cooperation is individually irrational.

• eg., The dominant strategy: advertise

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Classic Prisoners’ DilemmaRocky’s strategies

ConfessDeny

Ginger’sstrategies

Confess

5 yearsPrison

5 yearsPrison

7 yearsPrison

Go free

1 yearPrison

1 yearPrison

7 yearsPrison

Go freeDenyDominant strategy: confess, even though they would both be better off if they both kept their mouths shut.

Dominant Strategy Equilibrium

Definition: A Dominant Strategy Equilibrium occurs when each player uses a dominant strategy.

HondaHonda

ToyotaToyota

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Dominated Strategy

Definition: A player has a dominated strategy when the player has another strategy that gives it a higher payoff no matter what the other player does.

HondaHonda

ToyotaToyota

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Dominant or Dominated Strategy

Why look for dominant or dominated strategies?

A dominant strategy equilibrium is particularly compelling as a "likely" outcome

Similarly, because dominated strategies are unlikely to be played, these strategies can be eliminated from consideration in more complex games. This can make solving the game easier.

Why look for dominant or dominated strategies?

A dominant strategy equilibrium is particularly compelling as a "likely" outcome

Similarly, because dominated strategies are unlikely to be played, these strategies can be eliminated from consideration in more complex games. This can make solving the game easier.

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HondaHonda

Dominated Strategy

ToyotaToyota

"Build Large" is dominated for each playerBy eliminating the dominated strategies, we can reduce the game matrix.

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Finding Nash Equilibrium Cases

1) Nash Equilibrium where Dominant Strategies overlap

2) Nash Equilibrium with one Dominant Strategy

3) Nash Equilibrium by eliminating Dominated Strategy

4) Nash Equilibrium through Best Responses

1) Nash Equilibrium where Dominant Strategies overlap

2) Nash Equilibrium with one Dominant Strategy

3) Nash Equilibrium by eliminating Dominated Strategy

4) Nash Equilibrium through Best Responses

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StudentStudent

Nash Equilibrium – Dominant Overlap

ProfessorProfessor

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StudentStudent

Nash Equilibrium – One Dominant

ProfessorProfessor

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StudentStudent

Nash Equilibrium – Eliminate Dominated

ProfessorProfessor

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StudentStudent

Nash Equilibrium – Best Responses

ProfessorProfessor

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Nash Equilibrium

• However it is found, a Nash Equilibrium ALWAYS occurs where Best Responses line up

• If Multiple Nash Equilibria exist, we can’t conclude WHICH outcome will occur, only the possible outcomes that can occur

• Also, it is often APPEARS that no Nash Equilibria exist:

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BarneyBarney

No Nash Equilibrium

FredFred

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Mixed StrategiesPure Strategy – A specific choice of a strategy from the player’s possible strategies in a game. (ie: Rock)

Mixed Strategy – A choice among two or more pure strategies according to pre-specified probabilities. (ie: Rock, Paper or Scissors each 1/3rd of the time)

If Pure Strategies can’t produce a Nash Equilibrium, Mixed Strategies can:

If both players randomize each choice 1/3rd of the time, nether have an incentive to deviate.